DOCSLIB.ORG

Slotine • Li APPLIED NONLINEAR CONTROL

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Slotine • Li APPLIED NONLINEAR CONTROL Slotine • Li APPLIED NONLINEAR CONTROL ! i APPLIED NONLINEAR CONTROL Jean-Jacques E Slotine Weiping Li Applied Nonlinear Control JEAN-JACQUES E. SLOTINE Massachusetts Institute of Technology WEIPING LI Massachusetts Institute of Technology' Prentice Hall Englewood Cliffs, New Jersey 07632 Library of Congress Cataloging-in-Publication Data Slotine, J.-J. E. (Jean-Jacques E.) Applied nonlinear control / Jean-Jacques E. Slotine, Weiping Li p. cm. Includes bibliographical references. ISBN 0-13-040890-5 1, Nonlinear control theory. I. Li, Weiping. II. Title. QA402.35.S56 1991 90-33365 629.8'312-dc20 C1P Editorial/production supervision and interior design: JENNIFER WENZEL Cover design: KAREN STEPHENS Manufacturing Buyer: LORI BULWIN =^= © 1991 by Prentice-Hall, Inc. ^=&= A Division of Simon & Schuster T k Englewood Cliffs, New Jersey 07632 All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher. Printed in the United States of America 20 19 18 17 16 15 14 13 12 1] ISBN D-13-DHDfiTa-S Prentice-Hall International (UK) Limited, London Prentice-Hall of Australia Pty. Limited, Sydney Prentice-Hall Canada Inc., Toronto Prentice-Hail Hispanoamericana, S.A., Mexico Prentice-Hall of India Private Limited, New Delhi Prentice-Hall of Japan, Inc., Tokyo Simon & Schuster Asia Pte. Ltd., Singapore Editora Prentice-Hall do Brasil, Ltda., Rio de Janeiro To Our Parents Contents Preface xi 1. Introduction 1 1.1 Why Nonlinear Control ? 1 1.2 Nonlinear System Behavior 4 1.3 An Overview of the Book 12 1.4 Notes and References 13 Part I: Nonlinear Systems Analysis 14 Introduction to Part I 14 2. Phase Plane Analysis 17 2.1 Concepts of Phase Plane Analysis 18 2.1.1 Phase Portraits 18 2.1.2 Singular Points 20 2.1.3 Symmetry in Phase Plane Portraits 22 2.2 Constructing Phase Portraits 23 2.3 Determining Time from Phase Portraits 29 2.4 Phase Plane Analysis of Linear Systems 30 2.5 Phase Plane Analysis of Nonlinear Systems 32 2.6 Existence of Limit Cycles 36 2.7 Summary 38 2.8 Notes and References 38 2.9 Exercises 38 VI11 3. Fundamentals of Lyapunov Theory 40 3.1 Nonlinear Systems and Equilibrium Points 41 3.2 Concepts of Stability 47 3.3 Linearization and Local Stability 53 3.4 Lyapunov's Direct Method 57 3.4.1 Positive Definite Functions and Lyapunov Functions 58 3.4.2 Equilibrium Point Theorems 61 3.4.3 Invariant Set Theorems 68 3.5 System Analysis Based on Lyapunov's Direct Method 76 3.5.1 Lyapunov Analysis of Linear Time-Invariant Systems 77 3.5.2 Krasovskii's Method 83 3.5.3 The Variable Gradient Method 86 3.5.4 Physically Motivated Lyapunov Functions 88 3.5.5 Performance Analysis 91 3.6 Control Design Based on Lyapunov's Direct Method 94 3.7 Summary 95 3.8 Notes and References 96 3.9 Exercises 97 4. Advanced Stability Theory 100 4.1 Concepts of Stability for Non-Autonomous Systems 101 4.2 Lyapunov Analysis of Non-Autonomous Systems 105 4.2.1 Lyapunov's Direct Method for Non-Autonomous Systems 105 4.2.2 Lyapunov Analysis of Linear Time-Varying Systems 114 4.2.3 The Linearization Method for Non-Autonomous Systems 116 4.3 * Instability Theorems 117 4.4 * Existence of Lyapunov Functions 120 4.5 Lyapunov-Like Analysis Using Barbalat's Lemma 122 4.5.1 Asymptotic Properties of Functions and Their Derivatives 122 4.5.2 Barbalat's Lemma 123 4.6 Positive Linear Systems 126 4.6.1 PR and SPR Transfer Functions 126 4.6.2 The Kalman-Yakubovich Lemma 130 4.6.3 Positive Real Transfer Matrices 131 4.7 The Passivity Formalism 132 4.7.1 Block Combinations 132 4.7.2 Passivity in Linear Systems 137 IX 4.8 * Absolute Stability 142 4.9 * Establishing Boundedness of Signals 147 4.10 * Existence and Unicity of Solutions 151 4.11 Summary 153 4.12 Notes and References 153 4.13 Exercises 154 5. Describing Function Analysis 157 5.1 Describing Function Fundamentals 158 5.1.1 An Example of Describing Function Analysis 158 5.1.2 Applications Domain 162 5.1.3 Basic Assumptions 164 5.1.4 Basic Definitions 165 5.1.5 Computing Describing Functions 167 5.2 Common Nonlinearities In Control Systems 169 5.3 Describing Functions of Common Nonlinearities 172 5.4 Describing Function Analysis of Nonlinear Systems 179 5.4.1 The Nyquist Criterion and Its Extension 180 5.4.2 Existence of Limit Cycles 182 5.4.3 Stability of Limit Cycles 184 5.4.4 Reliability of Describing Function Analysis 186 5.5 Summary 187 5.6 Notes and References 188 5.7 Exercises 188 Part II: Nonlinear Control Systems Design 191 Introduction to Part II 191 6. Feedback Linearization 207 6.1 Intuitive Concepts 208 6.1.1 Feedback Linearization And The Canonical Form 208 6.1.2 Input-State Linearization 213 6.1.3 Input-Output Linearization 216 6.2 Mathematical Tools 229 6.3 Input-State Linearization of SISO Systems 236 6.4 Input-Output Linearization of SISO Systems 246 6.5 * Multi-Input Systems 266 6.6 Summary 270 6.7 Notes and References 271 6.8 Exercises 271 7. Sliding Control 276 7.1 Sliding Surfaces 277 7.1.1 A Notational Simplification 278 7.1.2 * Filippov's Construction of the Equivalent Dynamics 283 7.1.3 Perfect Performance - At a Price 285 7.1.4 Direct Implementations of Switching Control Laws 289 7.2 Continuous Approximations of Switching Control Laws 290 7.3 The Modeling/Performance Trade-Offs 301 7.4 * Multi-Input Systems 303 7.5 Summary 306 7.6 Notes and References 307 7.7 Exercises 307 8. Adaptive Control 311 8.1 Basic Concepts in Adaptive Control 312 8.1.1 Why Adaptive Control ? 312 8.1.2 What Is Adaptive Control ? 315 8.1.3 How To Design Adaptive Controllers ? 323 8.2 Adaptive Control of First-Order Systems 326 8.3 Adaptive Control of Linear Systems With Full State Feedback 335 8.4 Adaptive Control of Linear Systems With Output Feedback 339 8.4.1 Linear Systems With Relative Degree One 340 8.4.2 Linear Systems With Higher Relative Degree 346 8.5 Adaptive Control of Nonlinear Systems 350 8.6 Robustness of Adaptive Control Systems 353 8.7 * On-Line Parameter Estimation 358 8.7.1 Linear Parametrization Model 359 8.7.2 Prediction-Error-Based Estimation Methods 364 8.7.3 The Gradient Estimator 364 8.7.4 The Standard Least-Squares Estimator 370 8.7.5 Least-Squares With Exponential Forgetting 374 8.7.6 Bounded-Gain Forgetting 376 8.7.7 Concluding Remarks and Implementation Issues 381 1.8 Composite Adaptation 382 1.9 Summary 388 1.10 Notes and References 389 1.11 Exercises 389 9. Control of Multi-Input Physical Systems 392 9.1 Robotics as a Prototype 393 9.1.1 Position Control 394 9.1.2 Trajectory Control 397 9.2 Adaptive Robot Trajectory Control 403 9.2.1 The Basic Algorithm 404 9.2.2 * Composite Adaptive Trajectory Control 411 9.3 Putting Physics in Control 416 9.3.1 High-Frequency Unmodeled Dynamics 416 9.3.2 Conservative and Dissipative Dynamics 418 9.3.3 Robotics as a Metaphor 419 9.4 Spacecraft Control 422 9.4.1 The Spacecraft Model 422 9.4.2 Attitude Control 425 9.5 Summary 432 9.6 Notes and References 433 9.7 Exercises 433 BIBLIOGRAPHY 437 INDEX 459 Preface In recent years, the availability of powerful low-cost microprocessors has spurred great advances in the theory and applications of nonlinear control. In terms of theory, major strides have been made in the areas of feedback linearization, sliding control, and nonlinear adaptation techniques. In terms of applications, many practical nonlinear control systems have been developed, ranging from digital "fly-by-wire" flight control systems for aircraft, to "drive-by-wire" automobiles, to advanced robotic and space systems. As a result, the subject of nonlinear control is occupying an increasingly important place in automatic control engineering, and has become a necessary part of the fundamental background of control engineers. This book, based on a course developed at MIT, is intended as a textbook for senior and graduate students, and as a self-study book for practicing engineers. Its objective is to present the fundamental results of modern nonlinear control while keeping the mathematical complexity to a minimum, and to demonstrate their use and implications in the design of practical nonlinear control systems. Although a major motivation of this book is to detail the many recent developments in nonlinear control, classical techniques such as phase plane analysis and the describing function method are also treated, because of their continued practical importance. In order to achieve our fundamental objective, we have tried to bring the following features to this book: • Readability: Particular attention is paid to the readability of the book by carefully organizing the concepts, intuitively interpreting the major results, and selectively using the mathematical tools. The readers are only assumed to have had one introductory control course. No mathematical background beyond ordinary differential equations and elementary matrix algebra is required. For each new result, interpretation is emphasized rather than mathematics. For each major result, we try to ask and answer the following key questions: What does the result intuitively and physically mean? How can it be applied to practical problems? What is its relationship to other theorems? All major concepts and results are demonstrated by examples.
Load more

Recommended publications
  • Basic Nonlinear Control Systems - D
    CONTROL SYSTEMS, ROBOTICS, AND AUTOMATION – Vol. III – Basic Nonlinear Control Systems - D. P. Atherton BASIC NONLINEAR CONTROL SYSTEMS D. P. Atherton University of Sussex, UK Keywords: Nonlinearity, friction, saturation, relay, backlash, frequency response, describing function, limit cycle, stability, phase plane, subharmonics, chaos, synchronization, compensation. Contents 1. Introduction 2. Forms of nonlinearity 3. Structure and behavior 4. Stability 5. Aspects of design 6. Conclusions Glossary Bibliography Biographical Sketch Summary This section is written to introduce the reader to nonlinearity and its effect in control systems, and also to ‘set the stage’ for the subsequent articles. Nonlinearity is first defined and then some of the physical effects which cause nonlinear behavior are discussed. This is followed by some comments on the stability of nonlinear systems, the existence of limit cycles, and the unique behavioral aspects which nonlinear feedback systems may possess. Finally a few brief comments are made on designing controllers for nonlinear systems. 1. Introduction Linear systems have the important property that they satisfy the superposition principle. This leads to many important advantages in methods for their analysis. For example, in circuit theory when an RLC circuit has both d.c and a.c input voltages, the voltage or current elsewhereUNESCO in the circuit can be– found EOLSS by summing the results of separate analyses for the d.c. and a.c. inputs taken individually, also if the magnitude of the a.c. voltage is doubledSAMPLE then the a.c. voltages or CHAPTERScurrents elsewhere in the circuit will be doubled. Thus, mathematically a linear system with input x()t and output yt() satisfies the property that the output for an input ax12() t+ bx () t is ay12() t+ by () t , if y1()t and y2 ()t are the outputs in response to the inputs x1()t and x2 ()t , respectively, and a and b are constants.
    [Show full text]
  • Neural Control for Nonlinear Dynamic Systems
    Neural Control for Nonlinear Dynamic Systems Ssu-Hsin Yu Anuradha M. Annaswamy Department of Mechanical Engineering Department of Mechanical Engineering Massachusetts Institute of Technology Massachusetts Institute of Technology Cambridge, MA 02139 Cambridge, MA 02139 Email: [email protected] Email: [email protected] Abstract A neural network based approach is presented for controlling two distinct types of nonlinear systems. The first corresponds to nonlinear systems with parametric uncertainties where the parameters occur nonlinearly. The second corresponds to systems for which stabilizing control struc­ tures cannot be determined. The proposed neural controllers are shown to result in closed-loop system stability under certain conditions. 1 INTRODUCTION The problem that we address here is the control of general nonlinear dynamic systems in the presence of uncertainties. Suppose the nonlinear dynamic system is described as x= f(x , u , 0) , y = h(x, u, 0) where u denotes an external input, y is the output, x is the state, and 0 is the parameter which represents constant quantities in the system. The control objectives are to stabilize the system in the presence of disturbances and to ensure that reference trajectories can be tracked accurately, with minimum delay. While uncertainties can be classified in many different ways, we focus here on two scenarios. One occurs because the changes in the environment and operating conditions introduce uncertainties in the system parameter O. As a result, control objectives such as regulation and tracking, which may be realizable using a continuous function u = J'(x, 0) cannot be achieved since o is unknown. Another class of problems arises due to the complexity of the nonlinear function f.
    [Show full text]
  • Control Theory
    Control theory S. Simrock DESY, Hamburg, Germany Abstract In engineering and mathematics, control theory deals with the behaviour of dynamical systems. The desired output of a system is called the reference. When one or more output variables of a system need to follow a certain ref- erence over time, a controller manipulates the inputs to a system to obtain the desired effect on the output of the system. Rapid advances in digital system technology have radically altered the control design options. It has become routinely practicable to design very complicated digital controllers and to carry out the extensive calculations required for their design. These advances in im- plementation and design capability can be obtained at low cost because of the widespread availability of inexpensive and powerful digital processing plat- forms and high-speed analog IO devices. 1 Introduction The emphasis of this tutorial on control theory is on the design of digital controls to achieve good dy- namic response and small errors while using signals that are sampled in time and quantized in amplitude. Both transform (classical control) and state-space (modern control) methods are described and applied to illustrative examples. The transform methods emphasized are the root-locus method of Evans and fre- quency response. The state-space methods developed are the technique of pole assignment augmented by an estimator (observer) and optimal quadratic-loss control. The optimal control problems use the steady-state constant gain solution. Other topics covered are system identification and non-linear control. System identification is a general term to describe mathematical tools and algorithms that build dynamical models from measured data.
    [Show full text]
  • Fundamentals of Backstepping Control
    Appendix Fundamentals of Backstepping Control This Appendix provides the fundamentals of the recursive backstepping control method. The presented material is a summary of more detailed descriptions that may be found in [43, 49]. Lyapunov-based controller design may be systematically tack- led by a recursive design procedure called backstepping. Backstepping is suitable for strict-feedback systems that are also known as “lower triangular”. An example of a strict-feedback systems is: ˙ ξ1 = f1(ξ1) + g1(ξ1)ξ2 ˙ ξ2 = f2(ξ1,ξ2) + g2(ξ1,ξ2)ξ3 . (A.1) ˙ ξr−1 = fr−1(ξ1,ξ2,...,ξr−1) + gr−1(ξ1,ξ2,...,ξr−1)ξr ˙ ξr = fr (ξ1,ξ2,...,ξr ) + gr (ξ1,ξ2,...,ξr )u where ξ1,...,ξr ∈ R, u ∈ R is the control input and fi , gi for i = 1,...,r are known functions. A typical feedback linearization approach in most cases leads to cancellation of useful nonlinearities. The backstepping design exhibits more flexi- bility compared to feedback linearization since it does not require that the resulting input–output dynamics be linear. Cancellation of potentially useful nonlinearities can be avoided resulting in less complex controllers. The main idea is to use some of the state variables of (A.1) as “virtual controls” or “pseudo controls”, and de- pending on the dynamics of each state, design intermediate control laws. The back- stepping design is a recursive procedure where a Lyapunov function is derived for the entire system. The recursive procedure can be easily expanded from the nominal case of a system augmented by an integrator. This is also referred to as integra- tor backstepping.
    [Show full text]
  • Nonlinear Control - an Overview
    1 NONLINEAR CONTROL - AN OVERVIEW Fernando Lobo Pereira, fl[email protected] Porto University, Portugal Nonlinear Control, 2008, FEUP References 1 M. Vidyasagar, Nonlinear Systems Analysis, Prentice-Hall, Englewood Cli®s, NJ, 1978. 2 A. Isidori, Nonlinear Control Systems, Springer-Verlag, 199X. 3 E. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, Springer-Verlag, 1998. 4 C. T. Chen, Introduction to Linear Systems Theory, Van Nostrand Reinhold Company, New York, NY, 1970. 5 C. A. Desoer, Notes for a Second Course on Linear Systems, Holt Rhinehart, Winston, 1970. 6 K. Ogata, State Space Analysis of Control Systems, Prentice-Hall, Englewood Cli®s, NJ, 1965. 7 J. M. Carvalho, Dynamical Systems and Automatic Control, Prentice-Hall, New York, NY, 1994. Fernando Lobo Pereira, fl[email protected] 1 Nonlinear Control, 2008, FEUP Introduction Problem description - system components, objectives and control issues. Example 1: telescope mirrors control PICTURE Most of real systems involve nonlinearities in one way or another. However, many concepts for linear systems play an important role since some of the techniques to deal with nonlinear systems are: ² Change of variable in phase space so that the resulting system is linear. ² Nonlinearities may be cancelled by picking up appropriate feedback control laws. ² Pervasive analogy ... Fernando Lobo Pereira, fl[email protected] 2 Nonlinear Control, 2008, FEUP Introduction Example 2: Pendulum Modeling PICTURE Dynamics (Newton's law for rotating objects): mθÄ(t) + mgsin(θ(t)) = u(t) State variables: θ; θ_ Control/Input variable: u (external applied torque) Let us assume m = 1; l = 1; g = 1. Fernando Lobo Pereira, fl[email protected] 3 Nonlinear Control, 2008, FEUP Introduction Control Design ½ (0; 0) stable equilibrium Stationary positions: (1) (¼; 0) unstable equilibrium.
    [Show full text]
  • Modeling, Control and Simulation of Control-Affine Nonlinear Systems with State-Dependent Transfer Functions
    New Jersey Institute of Technology Digital Commons @ NJIT Dissertations Electronic Theses and Dissertations Spring 5-31-2014 Modeling, control and simulation of control-affine nonlinear systems with state-dependent transfer functions Roger Kobla Kwadzogah New Jersey Institute of Technology Follow this and additional works at: https://digitalcommons.njit.edu/dissertations Part of the Electrical and Electronics Commons Recommended Citation Kwadzogah, Roger Kobla, "Modeling, control and simulation of control-affine nonlinear systems with state-dependent transfer functions" (2014). Dissertations. 164. https://digitalcommons.njit.edu/dissertations/164 This Dissertation is brought to you for free and open access by the Electronic Theses and Dissertations at Digital Commons @ NJIT. It has been accepted for inclusion in Dissertations by an authorized administrator of Digital Commons @ NJIT. For more information, please contact [email protected]. Copyright Warning & Restrictions The copyright law of the United States (Title 17, United States Code) governs the making of photocopies or other reproductions of copyrighted material. Under certain conditions specified in the law, libraries and archives are authorized to furnish a photocopy or other reproduction. One of these specified conditions is that the photocopy or reproduction is not to be “used for any purpose other than private study, scholarship, or research.” If a, user makes a request for, or later uses, a photocopy or reproduction for purposes in excess of “fair use” that user may be
    [Show full text]
  • Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles
    Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles by Reza Olfati-Saber Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering and Computer Science at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY February 2001 © Massachusetts Institute of Technology 2001. All rights reserved. Author ................................................... Department of Electrical Engineering and Computer Science January 15, 2000 C ertified by.......................................................... Alexandre Megretski, Associate Professor of Electrical Engineering Thesis Supervisor Accepted by............................................ Arthur C. Smith Chairman, Department Committee on Graduate Students Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles by Reza Olfati-Saber Submitted to the Department of Electrical Engineering and Computer Science on January 15, 2000, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering and Computer Science Abstract This thesis is devoted to nonlinear control, reduction, and classification of underac- tuated mechanical systems. Underactuated systems are mechanical control systems with fewer controls than the number of configuration variables. Control of underactu- ated systems is currently an active field of research due to their broad applications in Robotics, Aerospace Vehicles, and Marine Vehicles. The examples of underactuated systems include flexible-link robots, nobile robots, walking robots, robots on mo- bile platforms, cars, locomotive systems, snake-type and swimming robots, acrobatic robots, aircraft, spacecraft, helicopters, satellites, surface vessels, and underwater ve- hicles. Based on recent surveys, control of general underactuated systems is a major open problem. Almost all real-life mechanical systems possess kinetic symmetry properties, i.e.
    [Show full text]
  • Nonlinear Control Via Input-Output Feedback Linearization of a Robot Manipulator
    Advances in Science, Technology and Engineering Systems Journal Vol. 3, No. 5, 374-381(2018) ASTESJ www.astesj.com ISSN: 2415-6698 NonLinear Control via Input-Output Feedback Linearization of a Robot Manipulator Wafa Ghozlane*, Jilani Knani Laboratory of Automatic Research LA.R.A, Department of Electrical Engineering, National School of Engineers of Tunis ENIT, University of Tunis ElManar, 1002 Tunis, Tunisia. A R T I C L E I N F O A B S T R A C T Article history: This paper presents the input-output feedback linearization and decoupling algorithm for Received: 08 September, 2018 control of nonlinear Multi-input Multi-output MIMO systems. The studied analysis was Accepted: 13 October, 2018 motivated through its application to a robot manipulator with six degrees of freedom. The Online: 18 October, 2018 nonlinear MIMO system was transformed into six independent single-input single-output SISO linear local systems. We added PD linear controller to each subsystem for purposes Keywords: of stabilization and tracking reference trajectories, the obtained results in different Input-Output Feedback simulations shown that this technique has been successfully implemented. linearization Nonlinear system Robot manipulator MIMO nonlinear system SISO linear systems PD controller 1. Introduction the angular position of each joint of this robot arm for stabilization and tracking purposes. The obtained results in different In recent years, Feedback linearization has been attracted a simulations shown the efficiency of the derived approach [7]. great deal of interesting research. It's an approach designed to the nonlinear control systems, which based on the idea of This paper is organized as follows: It is divided into five transforming nonlinear dynamics into a linear form.
    [Show full text]
  • Design of Terminal Sliding Mode Controllers for Disturbed Non-Linear Systems Described by Matrix Differential Equations of the Second and First Orders
    applied sciences Article Design of Terminal Sliding Mode Controllers for Disturbed Non-Linear Systems Described by Matrix Differential Equations of the Second and First Orders Paweł Skruch * and Marek Długosz Department of Automatic Control and Robotics, AGH University of Science and Technology, al. A. Mickiewicza 30, 30-059 Krakow, Poland; [email protected] * Correspondence: [email protected] Received: 2 February 2019; Accepted: 1 June 2019; Published: 6 June 2019 Abstract: This paper describes a design scheme for terminal sliding mode controllers of certain types of non-linear dynamical systems. Two classes of such systems are considered: the dynamic behavior of the first class of systems is described by non-linear second-order matrix differential equations, and the other class is described by non-linear first-order matrix differential equations. These two classes of non-linear systems are not completely disjointed, and are, therefore, investigated together; however, they are certainly not equivalent. In both cases, the systems experience unknown disturbances which are considered bounded. Sliding surfaces are defined by equations combining the state of the system and the expected trajectory. The control laws are drawn to force the system trajectory from an initial condition to the defined sliding surface in finite time. After reaching the sliding surface, the system trajectory remains on it. The effectiveness of the approaches proposed is verified by a few computer simulation examples. Keywords: non-linear dynamical system; disturbance; sliding mode control; sliding surface 1. Introduction 1.1. Motivation Stabilization of non-linear systems finds applications in many areas of engineering, particularly in the fields of mechanics, robotics, electronics, and so on [1,2].
    [Show full text]
  • Neural Networks for Control
    Neural Networks for Control Martin T. Hagan School of Electrical & Computer Engineering Oklahoma State University [email protected] Howard B. Demuth Electrical Engineering Department University of Idaho [email protected] Abstract the inverse of a system we are trying to control, in which case the neural network can be used to imple- The purpose of this tutorial is to provide a quick ment the controller. At the end of this tutorial we overview of neural networks and to explain how they will present several control architectures demon- can be used in control systems. We introduce the strating a variety of uses for function approximator multilayer perceptron neural network and describe neural networks. how it can be used for function approximation. The backpropagation algorithm (including its variations) Output is the principal procedure for training multilayer Unknown perceptrons; it is briefly described here. Care must Function - be taken, when training perceptron networks, to en- Input Error sure that they do not overfit the training data and then fail to generalize well in new situations. Several Neural + techniques for improving generalization are dis- Network cused. The tutorial also presents several control ar- Predicted Adaptation chitectures, such as model reference adaptive Output control, model predictive control, and internal model control, in which multilayer perceptron neural net- works can be used as basic building blocks. Figure 1 Neural Network as Function Approximator In the next section we will present the multilayer 1. Introduction perceptron neural network, and will demonstrate In this tutorial we want to give a brief introduction how it can be used as a function approximator.
    [Show full text]
  • Sliding Mode Control of Constrained Nonlinear Systems
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Archivio istituzionale della ricerca - Politecnico di Milano 1 Sliding Mode Control of Constrained Nonlinear Systems chattering reduction is not the only possible purpose of HOSM, since this latter allows the use of a relative degree greater than one between Gian Paolo Incremona, Matteo Rubagotti, and Antonella Ferrara the discontinuous control input and the sliding variable. In the first-order SMC formulation, input saturations are imme- diately satisfied if the control variable switches between values Abstract—This technical note introduces the design of sliding mode control algorithms for nonlinear systems in the presence of hard inequal- that are inside the imposed boundaries. When HOSM is used for ity constraints on both control and state variables. Relying on general chattering reduction, specific solutions have been proposed to achieve results on minimum-time higher-order sliding mode for unconstrained the satisfaction of saturation constraints (see, e.g., [19], [20] for the systems, a general order control law is formulated to robustly steer the second-order case). On the other hand, the possible presence of state state to the origin, while satisfying all the imposed constraints. Results on minimum-time convergence to the sliding manifold, as well as on the constraints is usually not taken into account in SMC formulations. maximization of the domain of attraction, are analytically proved for Recently, few solutions have been proposed in order to merge SMC the first-order and second-order sliding mode cases. A general result is and MPC, and combine the constraints satisfaction property of MPC presented regarding the domain of attraction in the general order case, with the robustness of SMC [21]–[25].
    [Show full text]
  • Designing Intelligent MIMO Nonlinear Controller Based on Fuzzy Cognitive Map Method for Energy Reduction of the Buildings
    energies Article Designing Intelligent MIMO Nonlinear Controller Based on Fuzzy Cognitive Map Method for Energy Reduction of the Buildings Farinaz Behrooz 1,* , Rubiyah Yusof 1,2, Norman Mariun 3,4, Uswah Khairuddin 1,2 and Zool Hilmi Ismail 1,2 1 Malaysia -Japan International Institute of Technology (MJIIT), Universiti Teknologi Malaysia, Kuala Lumpur 54100, Malaysia 2 Centre for Artificial Intelligence and Robotics (CAIRO), Universiti Teknologi Malaysia, Kuala Lumpur 54100, Malaysia 3 Department of Electrical and Electronic Engineering, Faculty of Engineering, University Putra Malaysia, Serdang 43400, Selangor, Malaysia 4 Centre for Advanced Power and Energy Research, Faculty of Engineering, University Putra Malaysia, Serdang 43400, Selangor, Malaysia * Correspondence: [email protected]; Tel.: +60-10-4315144 Received: 27 May 2019; Accepted: 9 July 2019; Published: 16 July 2019 Abstract: Designing a suitable controller for air-conditioning systems to reduce energy consumption and simultaneously meet the requirements of the system is very challenging. Important factors such as stability and performance of the designed controllers should be investigated to ensure the effectiveness of these controllers. In this article, the stability and performance of the fuzzy cognitive map (FCM) controller are investigated. The FCM method is used to control the direct expansion air conditioning system (DX A/C). The FCM controller has the ability to do online learning, and can achieve fast convergence thanks to its simple mathematical computation. The stability analysis of the controller was conducted using both fuzzy bidirectional associative memories (FBAMs) and the Lyapunov function. The performances of the controller were tested based on its ability for reference tracking and disturbance rejection. On the basis of the stability analysis using FBAMS and Lyapunov functions, the system is globally stable.
    [Show full text]